Symmetry of the Friedmann–Lemaître–Robertson–Walker universe
Abstract
We present a detailed analysis of the symmetries and physical dynamics of the Friedmann–Lemaître–Robertson–Walker (FLRW) universe, highlighting its foundational role in modern cosmology. By explicitly solving the Killing equations, we derive the six spatial isometries — three translations and three rotations — characteristic of the maximal symmetry inherent to homogeneous and isotropic spacetimes. Focusing on two representative scale factor scenarios, hyperbolic and exponential expansions, we investigate the resulting energy–momentum dynamics. In both cases, we find that classical energy conditions are violated during cosmic expansion, suggesting the need for either exotic forms of matter or extensions to general relativity to consistently describe the observed universe.